Development lenght of FRP straight rebars.pdf

Development length of FRP straight rebars

E. Cosenza, G. Manfredi, R. Realfonzo*

Department of Structural Analysis and Design, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy

Received 25 June 2001; revised 8 July 2002; accepted 13 July 2002

Abstract

In recent years, some attempts have been performed to extend general design rules reported in the codes for steel reinforced concrete to

Fiber Reinforced Polymer (FRP) materials; this is the case of relationships adopted in the evaluation of the development length clearly

derived by extension of the formulations used for steel rebars. However, such relationships seem to be inappropriate for FRP reinforcing bars:

in fact, experimental test results have shown that bond behaviour of FRP bars is different from that observed in case of deformed steel ones.

As a consequence, a new procedure for the evaluation of development length based on an analytical approach is needed in order to directly

account for the actual bond-slip constitutive law as obtained by experimental tests on different types of FRP reinforcing bars.

An analytical solution of the problem of a FRP rebar embedded in a concrete block and pulled-out by means of a tensile force applied on

the free end is presented herein. Such solution leads to an exact evaluation of the development length when splitting failure is prevented.

Finally, based on the analytical approach, a limit state design procedure is suggested to evaluate the development length.q 2002 ElsevierScience Ltd. All rights reserved.

Keywords: FRP reinforcing bar

1. Introduction

From design point of view, the study of concrete

structures reinforced using FRP reinforcing bars has been

initially developed by extending the wide body of

information gathered in a century of use of steel reinforced

concrete. Studies have been often carried out by comparing

performances obtained by using steel or FRP reinforcing

bars; moreover, the manufacturing technologies have been

oriented to fabricate composite bars which are similar, in

shape and dimensions, to those made of deformed steel.

One of the critical aspects of structural behaviour is the

development of an adequate bond behaviour; a number of

tests have been performed by several authors on FRP

reinforcing bars in order to study their bond performance

and to compare such bond properties with those evidenced

by deformed steel bars. On this topic, three state-of-art

reports have been recently published by Cosenza et al. [8],

Tepfers [21] and fib Task Group 5.2 [12].

From the experimental results, it was concluded that

bond between FRP reinforcement and concrete is controlled

by several factors such as the mechanical and geometrical

properties of bars and the compressive strength of concrete.

In particular, the interaction phenomena are governed by

shear strength and deformability of ribs, which are

remarkably lower than those of steel bars; this leads to

increased slips between rebars and concrete and different

failure mechanisms [13,15,16]. Mechanical properties of

resin, which the matrix is made of, have a remarkable

influence on the interaction behaviour since they strongly

affect strength and deformability of ribs and indentations

located on the outer surface.

Therefore, by comparing FRP and steel rebars, it has been

noted that the differences in bond behaviour are due to some

properties of FRP reinforcing bars; this underlines the

inadequacy of extending the design rules for steel reinforced

concrete to FRP reinforcing bars. Therefore, a critical review

of the design methodology is needed in order to introduce this

new type of bars as reinforcement for concrete structures.

Despite the above remarks, some attempts have been

performed in order to extend to such new materials, with

some minor modifications, the general design rules reported

in the codes traditionally used for steel reinforced concrete;

this is the case of the relationships adopted in the evaluation

of the embedment length clearly derived by extension of the

formulation used for steel bars [3,14].

1359-8368/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S1 35 9 -8 36 8 (0 2) 00 0 51 -3

Composites: Part B 33 (2002) 493504

www.elsevier.com/locate/compositesb

* Corresponding author. Tel.: 39-81-7683485; fax: 39-81-7683406.E-mail address: [email protected] (R. Realfonzo).

This paper analyses the problem of evaluating the

development length of deformed FRP reinforcing bars and

presents a new approach based on an analytical formulation.

Such a procedure is based on the analytical study of the

problem of a rebar embedded in a concrete block and

pulled-out by means of a tensile force applied at one end. In

order to integrate the differential equation that governs such

problem, the definition of a suitable bond-slip constitutive

relationship is needed.

Recently, Pecce et al. [18] proposed a numerical procedure

for evaluating the constitutive bond-slip relationship. Based

on the experimental bond test results, such procedure is able

to identify values of the parameters which define the

modified version [8] of the Eligehausen, Popov and Bertero

relationship [10]. In this way, a suitable constitutive ts lawwas proposed in the case of a recently introduced GFRP

reinforcing bars (C-Bare by Marshall Inc.); however, theabove-mentioned numerical procedure could allow to derive

a constitutive relationship for any type of bars.

The presented analytical approach has been developed by

considering the Eligehausen et al., modified law as bond-

slip (ts ) relationship; a numerical example is carried outin case of C-Bare using the constitutive ts law suggestedby Pecce et al. [18].

Finally, based on the analytical approach, a design

procedure is proposed to evaluate the development length.

The method, which represents a first proposal, takes into

account the actual bond-slip constitutive laws of the different

FRP rebars and introduces some different safety factors.

2. Evaluation of the basic development length

Under the assumption of constant distribution of bond

stresses t, the problem of a diameter f reinforcing bar

embedded in a concrete block for a length Ld and subjected

to a tensile force T is governed by the following equilibrium

equation:

T Ldptf 1If Ab is the rebar area and ff the tensile stress, the tensile

force T can be written as:

T Abff 2From Eqs. (1) and (2), it follows:

Ld Abffptf 3

or, alternatively:

Ld fff4t

4For deformed steel bars, it has been found that bond strength

tm is a linear function of the square root of the compressiveconcrete strength f 0c [10]:

tm kf 0c

p5

where k is a constant.

Therefore, from Eqs. (3) and (5):

Ld Abffpkf

f 0c

p 6and setting K pkf; it follows:

Ld AbffK

f 0c

p 7Eq. (7) represents the well known basic development

length, generally indicated with Ldb; K depends on the

relationship between the bond strength and the compressive

concrete strength and on the bar diameter.

In case of #3 to #11 deformed steel reinforcing bars, the

Nomenclature

Ab transverse section area of the bar

cb bar perimeter

f bar diameterE elastic modulus of the FRP rebar

Ef elastic modulus of steel rebars

f 0ck characteristic compressive strength of concretefb0d design bond strength

fc compressive concrete strength

f 0c compressive concrete strengthfd design tensile strength of the FRP rebar

ff tensile stress of the FRP rebar

ft tensile concrete strength

fu tensile strength of the bar

fyf yielding strength of steel rebar in tension

Ld development length

Ldb basic development length

s slip

sm slip at peak bond strength

wlim allowable crack width

a,p parameters of the Eligehausen et al., modifiedbond-slip law (Eqs. (16) and (17))

1 tensile straingE safety factor that affects the elastic modulusgg global safety factorgm material safety factors tensile stresssu tensile strength of the FRP barsuk characteristic value of tensile strength of the

FRP rebar

syk characteristic yielding strength of steel rebarst bond stresstm maximum bond strength

E. Cosenza et al. / Composites: Part B 33 (2002) 493504494

ACI 318-89 [1] assumed a value of K equal to 25 thus

leading to the following expression of Ldb:

Ldb 0:04Abfyf

f 0cp 8

where fyf and f0c are the yielding strength of steel bars and

compressive concrete strength, respectively (psi), and Ab, is

the rebar area (in.2).

According to ACI 318-89, the development length Ld is

provided by:

Ld Y

fi

Ldb 9

whereQ

fi indicates the product of some modification

factors that take into account the influence on bond of some

key parameters (i.e. cover, spacing, transverse reinforce-

ment).

Some modifications of Eq. (8) have been subsequently

reported by ACI 318-95 [2].

In order to extend Eq. (7) to FRP reinforcing bars, several

investigators have attempted to evaluate experimentally

values of K for different types of FRP bars [5,9,11,19].

Furthermore, based on experimental results, in case of

FRP reinforcing bars, some authors proposed simplified

expressions of Ldb; these expressions, clearly design

oriented, are not suitable for all types of FRP reinforcing

bars, because they are practically appropriate only for the

selected bars. An example is given by [6,7]:

Ld 20f 10Recently, formulations for evaluating the basic develop-

ment length have been proposed in new codes for design

of concrete structures reinforced with FRP bars. This is

the case of the Japan Society of Civil Engineers (JSCE)

Design Code [14] and of the ACI Committee 440 Guide

[3].

In the case of JSCE code, Ldb is clearly derived from

Eq. (4):

Ldb a1 ffd4fb0d

11

where fd is the design tensile strength, fb0d, the design bond

strength and a1 is a coefficient less than 1.The Japanese code states that the basic development

length shall not be taken less than 20 times the bar diameter.

The ACI Committee 440, for failure controlled by

pullout, proposes (in SI units):

Ldb fffu18:5

12

where Ldb is in mm, ffu (ultimate design strength of FRP

reinforcing bars) in MPa and f is in mm.In the ACI guide, two modification factors greater than 1

are considered in order to prevent splitting of concrete and

to evaluate the development length of top bars.

The above relationships seem to be unsuitable for FRP

reinforcing bars since they have been derived by adopting

a linear relationship between bond strength and the square

root of the compressive concrete strength f 0c: Severalinvestigators have shown that such a relationship does not

hold true for FRP reinforcing bars [6,13,15,16].

Therefore, the development of a new procedure to

evaluate the development length of FRP bars, based on an

analytical approach, is needed. It should take into account

the actual bond-slip constitutive law, as obtained by

experimental tests.

3. The problem of the rebar pull-out

The problem of a rebar embedded in a concrete block and

pulled-out by means of an applied tensile force is analysed

in the following.

A closed form solution is obtained by adopting linear

elastic constitutive laws for the materials and the Eligehau-

sen et al., modified relationship [8,18] as bond-slip

constitutive law. Such an analytical solution allows to

obtain slip, normal stress and bond stress distributions along

the rebar (i.e. at a generic abscissa x ). Furthermore, such a

solution leads to an exact evaluation of the development

length.

The studied case is schematically shown in Fig. 1, where

the origin of the x-axis is in the free end of the bar.

The differential equation that governs the bond problem

[20] is obtained by considering:

the equilibrium of rebar:

pf2

4ds pft dx 13

a linear elastic behaviour for the rebar that, if thecontribution of concrete in tension is neglected, is given

by:

s E1 E dsdx

14

where E and f are the elastic modulus and the diameterof the rebar, respectively.

From Eqs. (13) and (14), the following differential

equation is obtained:

d2s

dx22

4

Eftx 0 15

The Eligehausen et al. modified model [8,18]shown in

Fig. 2is considered herein. Such a constitutive law is

given by:

(A) for s , sm; an ascending branch which is formallycoincident with the first branch of the Eligehausen, Popov

E. Cosenza et al. / Composites: Part B 33 (2002) 493504 495

and Bertero law [10]:

ts tm ssm

a16

(B) for sm , s , su a softening branch given by:

ts tm 1 p2 p ssm

17

where a is a coefficient which describes the ascendingbranch, p, a coefficient which defines the softening branch,

tm, the maximum bond strength, sm, the slip at peak bondstrength and su is the ultimate slip.

By using such a law when integrating Eq. (15), two cases,

A (s # sm) and B (s . sm) have to be separately considered.

3.1. Case A (s # sm)

Considering Eq. (16), it is possible to rewrite Eq. (15) as:

d2s

dx22

4tmEfsam

sa 0 18

By integrating Eq. (18) with the following boundary

conditions:

s0 0; dsdx

x0

10 0

i.e. considering a perfect anchorage of the bar, the following

solution is obtained:

sx 2tmEfsam

12 a21 a

" #1=12ax2=12a 19

that provides the trend of the slip s along the bar.

Trends of the bond stress t and of the tensile stress sare derived by considering Eqs. (13) and (14), respect-

ively:

tx pf4

ds

dx; sx E ds

dx20

Using the above-presented relationships, two characteristic

limit values can be derived from Eqs. (19) and (20):

the limit tensile stress in the bar s1; the limit development length lm.

The first value represents the stress in the bar

corresponding to a slip equal to sm:

s1 ssm 8E

f

tmsm1 a

s21

The value lm represents an upper bound of the development

length related to the ascending branch of the bond-slip law,

i.e. the development length that corresponds to a stress

applied to the rebar equal to s1. In fact, setting s sm in Eq.(19), the corresponding value of x represents lm:

lm Ef

2

smtm

1 a12 a2

s22

Furthermore, considering Eqs. (21) and (22), lm can be also

written as:

lm s1f4tm

1 a12 a

l0m 1 a12 a

23

where l0m is the development length evaluated for s s1and t constant tm.

Fig. 1. The studied cases.

Fig. 2. Eligehausen et al., modified ts law.

E. Cosenza et al. / Composites: Part B 33 (2002) 493504496

A reduction of the rebar elastic modulus E results in

an increment of slips s (Eq. (19)) and in a reduction of

the embedment length (Eq. (22)). Eq. (23) confirms that

lm is greater than l0m, since (1 a )/(1 2 a ) is greaterthan 1.

Eqs. (19), (20) and (22) lead to the following useful

relationships:

sxsm

xlm

pswhere ps 2

12 a24

sxs1

xlm

pswhere ps 1 a

12 a ps 2 1 25

txtm

xlm

ptwhere pt 2a

12 a ps 2 2 26

Eqs. (24)(26) provide simple expressions of s, s and t as afunction of x=lm: In particular, it can be noticed that:

for a 0, the slip s is a quadratic function of the abscissax, while the normal stress s is linear and the bond stress tassumes the constant value tm;

for a 1/3, sx is cubic, s(x) is parabolic and t(x) islinear.

For s , s1, the development length l can be evaluatedfrom Eq. (25) by setting x l; then, the followingexpression of l is obtained:

l lm ss1 12a=1a

27

For 0 , s # s1, Eq. (27) provides values of the length l lessthan the value lm derived from Eq. (23).

According to Eq. (23), another expression of l can be

easily derived from Eq. (27):

l l0 s1s 2a=1a 1 a

12 a28

where:

l0 l0s fs4tm

29

is the development length evaluated in case of t constant tm.

It has to be underlined that Eq. (29) can be obtained from

Eq. (28) by setting a 0, i.e. by assuming a rigid-plasticbond-slip constitutive law.

Finally, from Eq. (28), it can be seen that the value of the

development length l is greater than l0.

3.2. Case B (s . sm)

Considering Eq. (17), the differential Eq. (18) becomes:

d2s

dx2 4ptm

Efsms 41 ptm

Ef30

and integrating Eq. (30) with the following boundary

conditions:

slm sm; dsdx

xlm

1lm s1E

the function sx is obtained:sxsm

1p

1 p2 cosvx2 lm(

2p

1 a

ssinvx2 lm

9=; 31

where:

v 4ptmEfsm

s

1

lm

2p

1 a

s1 a12 a

Finally, by substituting Eq. (31) into Eq. (17), the

distribution law of bond stresses along the bar is given by:

txtm

cosvx2 lm2

2p

1 a

ssinvx2 lm 32

while, remembering Eq. (14), sx is provided by:sxs1

cosvx2 lm 1 a

2p

ssinvx2 lm 33

It is possible to demonstrate by Eq. (30) that the

development length l can be obtained by:

l lm 1 12 a2

2p1 a

sarcsin

At2

Atmax

s" #8